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Properties

Label 2-603-67.37-c1-0-22
Degree $2$
Conductor $603$
Sign $-0.456 - 0.889i$
Analytic cond. $4.81497$
Root an. cond. $2.19430$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 − 2.14i)2-s + (−2.05 + 3.56i)4-s − 2.75·5-s + (2.34 − 4.06i)7-s + 5.22·8-s + (3.40 + 5.89i)10-s + (2.47 − 4.28i)11-s + (0.0567 + 0.0983i)13-s − 11.6·14-s + (−2.34 − 4.06i)16-s + (−2.61 − 4.52i)17-s + (−0.209 − 0.363i)19-s + (5.66 − 9.80i)20-s − 12.2·22-s + (1.81 + 3.14i)23-s + ⋯
L(s)  = 1  + (−0.874 − 1.51i)2-s + (−1.02 + 1.78i)4-s − 1.23·5-s + (0.887 − 1.53i)7-s + 1.84·8-s + (1.07 + 1.86i)10-s + (0.745 − 1.29i)11-s + (0.0157 + 0.0272i)13-s − 3.10·14-s + (−0.586 − 1.01i)16-s + (−0.633 − 1.09i)17-s + (−0.0481 − 0.0833i)19-s + (1.26 − 2.19i)20-s − 2.60·22-s + (0.378 + 0.655i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $-0.456 - 0.889i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ -0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242165 + 0.396244i\)
\(L(\frac12)\) \(\approx\) \(0.242165 + 0.396244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 + (1.25 - 8.08i)T \)
good2 \( 1 + (1.23 + 2.14i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.75T + 5T^{2} \)
7 \( 1 + (-2.34 + 4.06i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.47 + 4.28i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0567 - 0.0983i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.61 + 4.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.209 + 0.363i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.81 - 3.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.19 - 5.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.69 - 6.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.63 + 9.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.42 - 7.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 1.11T + 43T^{2} \)
47 \( 1 + (-1.67 + 2.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.66T + 53T^{2} \)
59 \( 1 - 6.66T + 59T^{2} \)
61 \( 1 + (0.347 + 0.601i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-1.81 + 3.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.36 + 9.29i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.17 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.140 - 0.243i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.07T + 89T^{2} \)
97 \( 1 + (-9.46 - 16.3i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.44314449756660196166397591591, −9.165818295360907524049109392776, −8.583649537773435587214913539471, −7.65686111591378946124811218071, −7.02652619755486551866361908513, −4.88827228341339476886905748824, −3.80133777295120208124128193920, −3.36831427101553685581179106705, −1.46183473808533784256376027012, −0.37502764120638038752835289697, 1.92472926168204249913538076125, 4.13774855578873987410430548828, 5.03370519783623338058638686971, 6.07059640291436652150370415946, 6.96910140655185320111409440969, 7.85719735115453157621233081142, 8.451266661170538713641879156329, 9.038056304644310066207393539282, 9.994242846693810190712378482506, 11.25963740903138403690250126326

Graph of the $Z$-function along the critical line